Method for the production of ultrapure phosphorus by zone melting in a non-flammable environment, and the apparatus used in such method

ABSTRACT

In general, the present invention relates to the production of ultrapure phosphorus. In particular, the present invention relates to the method for the production of ultrapure phosphorus by zone melting in a Non-flammable environment and the equipments used in such method. The process of the present invention is clean, chemical free, fast, and energy efficient.

CROSS-REFERENCE TO RELATED APPLICATION

This application claims the benefit of U.S. Provisional Patent Application Ser. No. 61/282,449 filed on Feb. 12, 2010, hereby incorporated herein by reference in its entirety.

TECHNICAL FIELD

Generally, the present invention relates to the production of ultrapure phosphorus. In particular, the present invention relates to the method for the production of ultrapure phosphorus by zone melting in a Non-flammable environment and the apparatus used in such method.

BACKGROUND ART

Phosphorus acid is commonly used as food additive. It is commonly added in almost all kinds of soft drinks, juice, wine etc. As an important element for life forms, the phosphorus is mainly supplied for human being and animal in the form of the food intake.

For food industries, industrial phosphorus must be purified to lower the impurities concentration, especially the arsenic concentration. The high arsenic concentration can increase the risk of several types of cancer, such as skin and lung cancer. Industrial grade phosphorus contains arsenic of the concentration range from 50 to 140 ppm (some to even 300 ppm). Phosphorus and arsenic co-exist because they are both group V elements and share similar chemical properties, so it is difficult to separate them by simple chemical method. The arsenic concentration for the food grade phosphorus should be below 10 ppm, which can produce 85% phosphorus acid with the arsenic concentration of 2.8 ppm. According to both the Food and Drug Administration standard (FDA)^(1a) and the Food Chemicals Codex standard (FCC)^(1b), the arsenic concentration in food grade phosphoric acid (85%) should be below 3 ppm.

Moreover, ultrapure phosphorus is also greatly demanded in the electronics industry as well as the semiconductor industry.

Therefore, there is a continued need in the art to provide a process of phosphorus purification which is not energy costly and as clean as possible.

CONTENTS OF THE INVENTION

In one aspect of the present invention, there is provided a method of production of ultrapure phosphorus by zone melting in a non-flammable environment, comprising the steps of

-   -   a. Fixing a heater in a non-flammable environment;     -   b. Fixing a phosphorus bar through the heater;     -   c. Setting up a motor to control the heater moving along the         phosphorus bar;     -   d. Heating a part of the said bar to become a heat zone;     -   e. Moving the heater along the said bar in order to have the         heat zone moving with the heater;     -   f Repeat step e if necessary.

In a preferred aspect of the present invention, the length of the heater D, the radius of the phosphorus bar R, the temperature of the heater T_(h), and the moving velocity of the heater v should control the solutions of the following governing equations:

In r<R and 0<z<L,

${\frac{1}{M}\frac{\partial\varphi}{\partial t}} = {{\left( {K_{E} + {TK}_{S}} \right){\nabla^{2}\varphi}} - {\frac{1}{2}\left( {W_{E} + {TW}_{S}} \right){g^{\prime}(\varphi)}} + {\frac{L_{V}\left( {T - T_{M}} \right)}{T_{M}}{p^{\prime}(\varphi)}}}$ $\mspace{79mu} {{{c_{P}\frac{\partial T}{\partial t}} + {L_{V}{r^{\prime}(\varphi)}\frac{\partial\varphi}{\partial t}} + {\frac{1}{2}W_{E}{g^{\prime}(\varphi)}\frac{\partial\varphi}{\partial t}}} = {{\nabla{\cdot \left\lbrack {{k(\varphi)}{\nabla T}} \right\rbrack}} + {K_{E}\frac{\partial\varphi}{\partial t}{\nabla^{2}\varphi}}}}$

The boundary conditions are:

${{{k(\varphi)}\frac{\partial T}{\partial n}} = {h\left( {T_{h} - T} \right)}},{{{for}\mspace{14mu} {v \cdot t}} \leq z \leq {{v \cdot t} + D}}$ ${{{and}\mspace{14mu} r} = R},{{{k(\varphi)}\frac{\partial T}{\partial n}} = {h\left( {T_{0} - T} \right)}},{{{for}\mspace{14mu} z} < {v \cdot t}},{z > {{v \cdot t} + D}}$ ${{{and}\mspace{14mu} r} = R},{z = 0},{z = L},{\frac{\partial\varphi}{\partial n} = 0},{{{for}\mspace{14mu} r} = R},{z = 0},{z = {L.}}$

In these equations, φ(x,t) is an continuous function which labels the two phases at the position x and the time t, with φ=0 in the solid phase, φ=1 in the liquid phase and 0<φ<1 in the interfacial region. M is the mobility coefficient, K_(E) and K_(S) are interfacial energy coefficients, W_(E) and W_(S) are barrier heights for double well potential function g(φ)=φ²(1−φ)². p(φ)=φ²(3−2φ). L_(v), is the latent heat per unit volume and T_(M) is the melting point. T(x,t) is the temperature distribution at the position x and the time t. c_(p) is the heat capacity per unit volume, r(φ)=p(φ). k(φ) permits different thermal conductivities k_(L)=k(0) and k_(S)=k(1) in the bulk liquid and solid phases, respectively. These governing equations are solved on a cylindrical domain x=(z,r)ε[0,L]×[0,R]. L is the length of phosphorus cylinder and R is the radius. D is the length of the heater. z is the axial direction coordinate. r is the radial direction coordinate. n is the outer normal vector of the boundary, h is the heat transfer coefficient, T₀ is the room temperature and T_(h) is the temperature of the heater.

In another aspect of the present invention, there is provided an apparatus for the method of the production of ultrapure phosphorus as mentioned above, which comprises in a non-flammable environment:

-   -   a). a heater;     -   b). a phosphorus bar fixed through the heater;     -   c). a motor set up to control the heater moving along the         phosphorus bar.

DESCRIPTION OF FIGURES

FIG. 1: A schematic drawing illustrating the apparatus for the zone-melting purification of phosphorus according to one embodiment of the present invention.

FIG. 2: A schematic drawing showing the detail of the heater used in the apparatus according to one embodiment of the present invention.

FIG. 3: A schematic drawing of the experimental apparatus for sample analysis used in one embodiment of the present invention.

FIG. 4: A figure showing the moving characteristic of the fusion region and the heater according to one embodiment of the present invention, wherein temperature of the heater is 50° C., and the velocity of the heater is set at 0.1 mm/s.

FIG. 5: A figure showing the temperature over the whole phosphorus cylinder at r=0, r=R/2 and r=R while the heater moves as described in FIG. 4.

FIG. 6: A figure showing the temperature over the whole phosphorus cylinder of FIG. 9 in 2D wherein t=400 s

FIG. 7: A figure showing the moving characteristic of the fusion region and the heater according to one embodiment of the present invention, wherein temperature of the heater is 50° C., and the velocity of the heater is set at 0.1 mm/s.

FIG. 8: A figure showing the moving characteristic of the fusion region and the heater according to one embodiment of the present invention, wherein temperature of the heater is 50° C., and the velocity of the heater is set at 0.2 mm/s. FIG. 8( a) shows the left interface of the fusion region lags behind the left boundary of the heater; the temperature at different radii at t=250 s are shown in (b).

FIG. 9: A figure showing the moving characteristic of the fusion region and the heater according to one embodiment of the present invention, wherein temperature of the heater is 50° C., and the velocity of the heater is set at 0.05 mm/s. FIG. 9( a) shows the fusion region stays in the middle of the heater; the temperature at different radii at t=600 s are shown in (b).

FIG. 10: A figure showing the moving characteristic of the fusion region and the heater according to one embodiment of the present invention, wherein temperature of the heater is 50° C., and the velocity of the heater is set at 0.02 mm/s. FIG. 10( a) shows the fusion region stays in the middle of the heater; the temperature at different radii at t=2000 s are shown in (b).

FIG. 11: A figure showing the moving characteristic of the fusion region and the heater according to one embodiment of the present invention, wherein temperature of the heater is 50° C., and the velocity of the heater is set at 0.5 mm/s. FIG. 11( a) shows the left interface of the fusion region lags behind the left boundary of the heater; the temperature at different radii at t=150 s are shown in (b).

FIG. 12: A figure showing the moving characteristic of the fusion region and the heater at different time, wherein the temperature of the heater is 45° C., and the velocity of the heater is set at 0.05 mm/s. The width of the melting region is much narrower than the length of the heater.

FIG. 13: A figure showing the temperature over the whole phosphorus cylinder at r=0, r=R/2 and r=R while the heater moves as described in FIG. 12.

FIG. 14: A figure showing the moving characteristic of the fusion region and the heater at different time, wherein the temperature of the heater is 45° C., and the velocity of the heater is set at 0.02 mm/s.

FIG. 15: A figure showing the moving characteristic of the fusion region and the heater according to one embodiment of the present invention, wherein the temperature of the heater is 55° C., and the velocity of the heater is set at 0.05 mm/s. The temperature at different radii at t=400 s are shown in (b).

FIG. 16: A figure showing the moving characteristic of the fusion region and the heater according to one embodiment of the present invention, wherein the temperature of the heater is 60° C., and the velocity of the heater is set at 0.05 mm/s. The temperature at different radii at t=600 s are shown in (b)

FIG. 17: A figure showing the moving characteristic of the fusion region and the heater according to one embodiment of the present invention, wherein the temperature of the heater is 70° C., and the velocity of the heater is set at 0.02 mm/s. The temperature at different radii at t=1600 s are shown in (b).

FIG. 18: A figure showing the moving characteristic of the fusion region and the heater according to one embodiment of the present invention, wherein the temperature of the heater is 80° C., and the velocity of the heater is set at 0.02 mm/s. The temperature at different radii at t=1600 s are shown in (b).

FIG. 19: A figure showing the moving characteristic of the fusion region and the heater according to one embodiment of the present invention, wherein the temperature of the heater is 50° C., and the velocity of the heater is set at 0.1 mm/s, the length d of the heater is 1.5 cm, and the radius of phosphorus cylinder is 5 mm. The temperature at different radii at t=500 s are shown in (b).

FIG. 20: A figure showing the moving characteristic of the fusion region and the heater according to one embodiment of the present invention, wherein the temperature of the heater is 50° C., and the velocity of the heater is set at 0.1 mm/s, the length d of the heater is 1.0 cm, and the radius of phosphorus cylinder is 5 mm. The temperature at different radii at t=800 s are shown in (b). The core of the cylinder cannot be melted. It will keep this shape in (a) after t=800 s.

FIG. 21: A figure showing the moving characteristic of the fusion region and the heater according to one embodiment of the present invention, wherein the temperature of the heater is 50° C., and the velocity of the heater is set at 0.05 mm/s, the length d of the heater is 3.0 cm, and the radius of phosphorus cylinder is 1 cm. The temperature at different radii at t=1300 s are shown in (b).

FIG. 22: A figure showing the moving characteristic of the fusion region and the heater according to one embodiment of the present invention, wherein the temperature of the heater is 50° C., and the velocity of the heater is set at 0.05 mm/s, the length d of the heater is 2.0 cm, and the radius of phosphorus cylinder is 1 cm. The temperature at different radii at t=2000 s are shown in (b). The core of the cylinder cannot be melted. It will keep this shape in (a) after t=2000 s.

FIG. 23: A figure showing the moving characteristic of the fusion region and the heater according to one embodiment of the present invention, wherein the temperature of the heater is 50° C., and the velocity of the heater is set at 0.05 mm/s, the length d of the heater is 4.0 cm, and the radius of phosphorus cylinder is 1.5 cm. The temperature at different radii at t=1900s are shown in (b).

FIG. 24: A figure showing the moving characteristic of the fusion region and the heater according to one embodiment of the present invention, wherein the temperature of the heater is 50° C., and the velocity of the heater is set at 0.05 mm/s, the length d of the heater is 3.0 cm, and the radius of phosphorus cylinder is 1.5 cm. The temperature at different radii at t=2000 s are shown in (b). The core of the cylinder cannot be melted. It will keep this shape in (a) after t=2000 s.

FIG. 25: A figure showing the moving characteristic of the fusion region and the heater according to one embodiment of the present invention, wherein the temperature of the heater is 60° C., and the velocity of the heater is set at 0.05 mm/s, the length d of the heater is 10.0 cm, and the radius of phosphorus cylinder is 5.0 cm. The fusion region has been separated to two parts so that it cannot wipe over the whole cylinder.

FIG. 26: A figure showing the moving characteristic of the fusion region and the heater according to one embodiment of the present invention, wherein the temperature of the heater is 60° C., and the velocity of the heater is set at 0.02 mm/s, the length d of the heater is 10.0 cm, and the radius of phosphorus cylinder is 5.0 cm. The fusion region can wipe over the whole cylinder.

FIG. 27: A figure showing the moving characteristic of the fusion region and the heater according to one embodiment of the present invention, wherein the temperature of the heater is 60° C., and the velocity of the heater is set at 0.1 mm/s, the length d of the heater is 15.0 cm, and the radius of phosphorus cylinder is 5.0 cm. The fusion region has been separated to two parts so that it cannot wipe over the whole cylinder.

FIG. 28: A figure showing the moving characteristic of the fusion region and the heater according to one embodiment of the present invention, wherein the temperature of the heater is 60° C., and the velocity of the heater is set at 0.05 mm/s, the length d of the heater is 15.0 cm, and the radius of phosphorus cylinder is 5.0 cm. The fusion region can wipe over the whole cylinder.

FIG. 29: A figure showing the moving characteristic of the fusion region and the heater according to one embodiment of the present invention, wherein the temperature of the heater is 60° C., and the velocity of the heater is set at 0.1 mm/s, the length d of the heater is 25.0 cm, and the radius of phosphorus cylinder is 5.0 cm. The fusion region can wipe over the whole cylinder.

MODE OF CARRYING OUT THE INVENTION Definitions

Throughout the present description, the term “non-flammable environment” refers to any environment which could prevent the flame of phosphors. Preferably the method of the present invention is carried out in water.

The Purification Methods of Phosphorus in the Prior Art

Phosphorus has a melting point of 44.2° C. and is extremely flammable when exposed to air. Because it does not have conventional solvent, purification of phosphorus is very difficult.

In summary, there are the following purification methods in the prior art.

(1) Vacuum Distillation⁷

Excessive iodine (around 10,000 times in excess) is added to the white phosphorus by the stoichiometric amount. Iodine would then react with arsenic to form arsenic iodide. Arsenic iodide has a much higher boiling point than phosphorus and, thus, arsenic can be separated from phosphorus by this vacuum distillation method. Most of the arsenic can be separated from phosphorus through the vacuum distillation by the difference in boiling point. The disadvantage is that large amounts of iodine, which is corrosive and poisonous, would be produced by this process. This method is, therefore, very dangerous to operate.

(2) Liquid-Liquid Extraction⁸

Crude yellow phosphorus is first mixed with the oxidant, nitric acid. Most of the arsenic inside the phosphorus would react to form arsenic acid but only part of the phosphorus would react to form phosphoric acid. When the mixture is heated to inciting point or more phosphorus is melted, liquid phosphorus and nitric acid are separated into two layers by the difference of specific gravity. Most of the arsenic acid is dissolved into nitric acid due to different selectivity and thus separation of the arsenic can be done by extracting the liquid phosphorus. This method is not widely used because it uses nitric acid as a solvent, which is a strong acid and highly corrosive and highly oxidant.

(3) Fractional Distillation⁸

Impure phosphorus is continually passed through a heat exchanger and heated to vaporize a portion of the phosphorus at a temperature not above 200° C. The materials with large boiling point difference to vaporize phosphorus will be separated in a disengaging zone, while arsenic will be separated in a fractionating zone due to the vapour-liquid equilibrium, Distillation is capital and labour intensive and also leaves a distillation residue product that is much higher in arsenic and other heavy metals due to the concentration caused by the distillation.

(4) Mixing and Extraction¹⁰

The purification of yellow phosphorus can be done in this process by mixing liquid yellow phosphorus with active carbon and then separating the purified phosphorus from the spent active carbon containing adhering phosphorus. The spent active carbon is mixed with sufficient liquid phosphorus to form slurry which is pumped to a disposal zone. The disposal of the spent active carbon can cause several problems. For example, the active carbon spent gives rise to a useless and insignificant consumption of phosphorus.

(5) Phase Separation¹¹

Impure white phosphorus mixed with an aqueous suspension of a purifying amount of active charcoal can be used to separate phosphorus from the separation funnel due to different phases. The mixture of aqueous impurities is first separated into a settled phase consisting essentially of a spent active charcoal and phosphorus-containing phase and an aqueous phase. Then the phosphorus-containing phase and thus, the purified liquid white phosphorus can be separated from another separation. However, the production of the active charcoal causes much pollution as the acid is involved and heavy metal is produced.

(6) Oxidation¹²

The purification of crude yellow phosphorus can be treated by oxidizing those with mixed acid of sulphuric acid and nitric acid. The crude yellow phosphorus is first pretreated by alkali leaching with ammonium carbonate. Then, the reactant is oxidized with mixed acid of sulphuric acid and nitric acid. The purified phosphorus can be found after washing with hot water. The strong acid, sulphuric acid and nitric acid, which is highly corrosive, and strongly oxidizing agent is used as the oxidant. This can result in water pollution when disposing them into the sea. The loss of phosphorus in reaction forming phosphorus acid would also occur.

It can be seen that the methods summarized above all have significant disadvantages such as high energy consumption with hazardous operation-conditions (toxic solvent and vapor) and low yield.

Therefore, in one aspect of the present invention, the objects thereof are to find an improved process for the production of ultrapure phosphorus which is clean, chemical free, fast, and energy efficient. It is absolutely a zero pollution process¹. In particular, the present invention provides a method for the production of ultrapure phosphorus by zone melting in non-flammable environment.

THE APPARATUS AND METHODS USED IN THE PRESENT INVENTION

According to one embodiment of the present invention, the apparatus used for zone melting includes a row of ring-shaped heater that moves slowly along a tube containing the solid to be purified. The heater heats a narrow band of the material, above the melting point, form the heat zone. Then the heat zone moves along with the heater. After heater passes, the heat zone cools and freezes. The impurities tend to stay in the heat zone and are carried to one end of the tube.

The impurities which could be purified by the zone melting purification should satisfy two important facts, which are (1) the diffusion coefficients of the impurities in the liquid zone are several orders higher in magnitude than that in the solid region, and (2) the solubility of the impurities in the amorphous liquid is significantly higher than that in the crystal solid.

Therefore, the term “impurities” used in the present disclosure refers to any materials satisfying the above two requirements, preferably Arsenic.

The apparatus for the zone-melting purification of phosphorus according to one embodiment of the present invention is illustrated in FIG. 1 ¹.

Because phosphorus is very flammable in the air, all the operation must be in non-flammable environment, preferably in water. The apparatus used in the present invention could be put in a stainless steel tank which is a water container with certain volume, for example 600 mm length, 420 mm width, and 300 mm height.

FIG. 1 shows the apparatus for zone-melting purification that includes three fixed pulleys, a step motor, and three slide rails. There is a plate which could be plastic plate or iron plate, with one side connected with the rails and its opposite side connected with the heater by the junction, as well as the top side fixed on the string of the pulleys by a junction. When the motor starts to work, the string draws the plate to move along rails, and then the heater will move along the phosphorus bar at the same time. There are two plastic end blocks for fixing the phosphorus bar by two screws on a rail. The phosphorus bar should be fixed through the heater and between the two plastic end blocks. The tank should be filled with material which provides non-flammable environment. Preferably the tank is filled with water. The water level must be kept above the holder of the heater in order to make sure all the conduction can be done under the water safely.

Although the apparatus illustrated in FIG. 1 is assembled in horizontal direction, the persons skilled in the art might understand that they could also be in vertical direction.

FIG. 2 shows the design of the heater. According to one embodiment, the heater is made by an Aluminum cylinder wrapped by the copper wire which is embedded in a plastic block. The length of the heater should be longer than the diameter of the phosphorus bar. According to one embodiment of the present invention, the length of the heater is at least 2 times, preferably at least 3 times, more preferably at least 4 times, longer than the diameter of the phosphorus bar. In the middle of the Aluminum cylinder, there is a thermal couple which can control heater's temperature. The thermal couple could be any suitable type, for example T type, and the sensitivity of the temperature can be set till 0.1° C.

The phosphorus bar can be molded by a plastic mold. The plastic mold was made by high density polyethylene or Teflon tube with a round hole in the middle. The diameter of the hole is 10 mm.

In this invention, the phosphorus bar is purified as it was molded in a cylindrical bar. But, for the convenience of future production, the phosphorus can also be confined in a case which can be plastic (polyethylene or Teflon, or glass).

The detailed information of all the chemicals and machines which were used in the present disclosure are listed below in Table 1 and Table 2:

TABLE 1 Chemicals Item Source Purity Yellow Phosphorus Yunnan Dean 99.96%, As ≦100 ppm Phosphorus Chemical Co. Ltd. Nitric Acid RDH 30702 70% Campus code: 1129 2.5 L

TABLE 2 Machines and device Item Source Type ICP-OES PERKIN-ELMER OPTIMA 3000XL Phosphorus Tube Self-designed Self-designed Purification Machine Self-designed(see the Self-designed FIG. 1)

EXAMPLES Determine of the As Concentration

The As concentration in the solid phosphorus were determined before and after the purification. Any test methods could be used in the present invention for the determine of As concentration. Preferably ICP-OES (Inductively Coupled Plasma Optical Emission Spectrometry) was used for analyzing the trace concentration of Arsenic.

Generally, the procedure of ICP-OES includes the following steps.

1. Sample preparation: 1) 15 ml of HNO₃ (70%) was added in 250 ml of flat bottom boiling flask and weighed W₁. A schematic drawing showing the apparatus could be seen in FIG. 3. 2) A piece of phosphorus (around 1 g) was added into the flat bottom boiling flask afterwards and weighed W₂. The weight difference between above procedures is the weight of phosphorus. Thus, the weight of phosphorus W_(p)=W₂-W₁. The weight of phosphorus should be around 1 g. 3) Then the flat bottom boiling flask was connected with the Jacket condenser quickly. During the reaction, the phosphorus should be kept below the HNO₃ liquid layer in order to keep the phosphorus from reacting with the air. The cool water should be kept to flow in the jacketed space in the condenser during the reaction. Finally, the phosphorus will react with HNO₃ (70%) completely. 4) After the reaction, the Jacketed Condenser was rinsed with distilled water and then removed. 5) The sample solution was transferred from the flat bottom boiling flask to 100 ml (“S_(vol)”) volumetric flask. Then the sample solution was diluted by distilled water till the mark of the flask.

2. Making Standard Curve

The concentration of the standard arsenic solution is 1000 ppm. 0.2 ml and 0.5 ml of the standard arsenic solution were taken accordingly and diluted to 100 ml separately. So the concentrations of the standard solution that determines the standard curve should be 2 and 5 ppm.

3. Measure

The concentration of As(CAs) in the diluted sample solution can be directly measured by ICP-OES with the standard curve.

4. Calculation:

The reading of ICP result is C_(As) (mg/l) The weight of phosphorus is W_(p) (g) Sample volume is S_(vol) (1) The concentration of As in P is A (ppm) (mg/kg)

A=C_(As)×S_(vol)/W_(p)×1000 Recovery Test

In order to know if there are some problems with the method or machine, it is necessary to do the recovery test. The procedure is described below:

Prepare the sample as discussed before, before HNO₃ (70%) was added in 250 ml of FB boiling flask, 0.1 ml of 506 ppm standard Arsenic solution was added in the FB boiling flask first.

As can be seen from Table 3-7, all the recovery values are above 80%, so the concentration of Arsenic tested by this method are reliable. It is also able to get the concentration of Arsenic in this industrial phosphorus. From above results, it is clear that the initial concentration of Arsenic in the industrial solid phosphorus is between 52-56 ppm.

TABLE 3 Concentration of Arsenic in solid P Sample weight/ Sample volume/ The reading from Conc. of As in g ml ICP/mg/l solid P/mg/kg 0.5109 50 0.5350 52.35 0.8893 100 0.4660 52.40 0.8120 100 0.4480 55.17 0.8139 100 0.4434 54.47

TABLE 4 Results of recovery experiments Sample Sample + Spike Sample weight/ 1.1345 0.8243 g Sample volume/ 100 100 ml As spiked/ 0 50.6 μg Solution conc./ 0.599 0.875 mg/l As in solution/ 59.9 87.5 μg As in solid/ 52.80 — μg/g As from solid in the solution/ 59.9 43.52 μg As from spike/ — 43.98 μg Recovery — 86.91% %

TABLE 5 Results of recovery experiments Sample Sample + Spike Sample weight/ 1.1516 0.9260 g Sample volume/ 100 100 ml As spiked/ 0 50.6 μg Solution conc./ 0.645 0.975 mg/l As in solution/ 64.5 97.5 μg As in solid/ 56.0 — μg/g As from solid in the solution/ 64.5 51.86 μg As from spike/ — 45.64 μg Recovery — 90.2% %

TABLE 6 Results of recovery experiments Sample Sample + Spike Sample weight/ 0.7798 0.9232 g Sample volume/ 100 100 ml As spiked/ 0 50.6 μg Solution conc./ 0.426 0.937 mg/l As in solution/ 42.6 93.7 μg As in solid/ 54.6 — μg/g As from solid in the solution/ 42.6 50.4 μg As from spike/ — 43.3 μg Recovery — 85.6% %

TABLE 7 Results of recovery experiments Sample Sample + Spike Sample weight/ 1.1175 0.9855 g Sample volume/ 100 100 ml As spiked/ 0 50.6 μg Solution conc./ 0.588 0.964 mg/l As in solution/ 58.8 96.4 μg As in solid/ 52.6 — μg/g As from solid in the solution/ 58.8 51.8 μg As from spike/ — 44.6 μg Recovery — 88.1% %

Purification

In the experiments of the present invention, the step motor moving speed was controlled from 0.02 min/min to 0.1 mm/min, and the heater temperature range was set from 45° C. to 80° C.

TABLE 8 Concentration of Arsenic in solid P Sample weight/ Sample volume/ The reading from Conc. of As in g ml ICP/mg/l solid P/mg/kg 1.1345 100 0.599 52.8 1.1516 100 0.645 56.0 0.7798 100 0.426 54.6 1.1175 100 0.588 52.6

A phosphorus bar (Bar No. 1) was put inside the stainless steel tube. The temperature of the heater is set at 48° C.), and speed was set at 0.375 mm/minute. Run three times at this condition. Then the two samples from the middle of the bar were taken and test by ICP. The results are shown in table 9.

TABLE 9 Concentration of Arsenic after Purification. (Bar No. 1) Sample weight/ Sample volume/ The reading from Conc. of As in g ml ICP/mg/l solid P/mg/kg 0.9728 100 0.356 36.6 1.2323 100 0.415 33.6

In order to improve the purification, more running times have been tried. A new phosphorus bar (Bar No. 2) was put inside the stainless steel tube. The temperature of the heater is set at 46° C. this time, and speed was set at 0.375 mm/minute as last time. Run eight times at this condition. Then the two samples from the middle of the bar was taken and tested by ICP. The results are shown in table 10.

TABLE 10 Concentration of Arsenic after Purification. (Bar No. 2) Sample weight/ Sample volume/ The reading from Conc. of As in g ml ICP/mg/l solid P/mg/kg 0.9994 100 0.359 35.9 1.4192 100 0.494 34.8

A new phosphorus bar (Bar No. 3) was put inside the stainless steel tube. The speed was set at 0.375 mm/minute. Run three times at 46° C., two times at 48° C., one time at 47° C. Then, the Arsenic concentration was successfully reduced below 10 ppm. See Table 11.

TABLE 11 Concentration of Arsenic after Purification. (Bar No. 3) Sample weight/ Sample volume/ The reading from Conc. of As in g ml ICP/mg/l solid P/mg/kg 0.872 100 <0.081 <10

Mathematical Model and Numerical Simulation:

The inventor of the present invention has developed a mathematical model to simulate the above zone melting purification process. The mathematical model is based on a phase field model for solid-liquid phase transition^(14, 15, 16) and is described in the following:

$\begin{matrix} {{\frac{1}{M}\frac{\partial\varphi}{\partial t}} = {{\left( {K_{E} + {TK}_{S}} \right){\nabla^{2}\varphi}} - {\frac{1}{2}\left( {W_{E} + {TW}_{S}} \right){g^{\prime}(\varphi)}} + {\frac{L_{V}\left( {T - T_{M}} \right)}{T_{M}}{p^{\prime}(\varphi)}}}} & (1) \\ {{{c_{P}\frac{\partial T}{\partial t}} + {L_{V}{r^{\prime}(\varphi)}\frac{\partial\varphi}{\partial t}} + {\frac{1}{2}W_{E}{g^{\prime}(\varphi)}\frac{\partial\varphi}{\partial t}}} = {{\nabla{\cdot \left\lbrack {{k(\varphi)}{\nabla\; T}} \right\rbrack}} + {K_{E}\frac{\partial\varphi}{\partial t}{\nabla^{2}\varphi}}}} & (2) \end{matrix}$

The first equation describes the evolution of the two phases: solid and liquid phase. φ(x,t) is an unknown continuous function which labels the two phases at the position x and the time t, with φ=0 in the solid phase, φ=1 in the liquid phase and 0<φ<1 in the interfacial region. The left side describes the time evolution of phase function φ(x,t): M is the mobility coefficient. The force leading to the evolution is given on the right side. The first term describes the effect of the interfacial energy on the dynamics of φ(x,t): K_(E) and K_(S) are interfacial energy coefficients; the Laplace ∇²φ indicates the variation of φ(x,t) in the interface. The second term describes the effect of the potential energy: W_(E) and W_(S) are barrier heights for double well potential function g(φ)=φ²(1−φ)². K_(E), K_(S), W_(E) and W_(S), are phenomenological parameters and can be determined later. The third term describes the force due to the latent heat released in the phase transition: p(φ)=φ²(3−2φ) is an interpolating function that records the proportion of the latent heat released in an intermediate state, L_(V), is the latent heat per unit volume and T_(M) is the melting point.

The second equation, which is just the generalization of a regular heat equation, describes the evolution of the temperature T(x,t) at the position x and the time t. The first term on the left side describes the time evolution of the temperature: c_(p) is the heat capacity per unit volume. The second term describes the effect of phase transition on the dynamics of temperature: r(φ)=p(φ) has the same meaning of p(φ). The third term describes the effect of potential energy. The first term on the right side describes the heat conduction by Fourier's Law: k(φ) permits different thermal conductivities k_(L)=k(0) and k_(S)=k(1) in the bulk liquid and solid phases, respectively, and varies continuously in the interfacial region. The second term on the right side describes the effect of interfacial energy.

All the parameters interpreted here are fixed physical parameters of phosphorus except the unknown functions φ(x,t) and T(x,t).

This system of equations can be derived from the conservation laws for energy and entropy and the fact that the entropy production is positive^(16, 17, 18.)

Let {tilde over (x)}={tilde over (t)}/R, {tilde over (t)}=t/(R²/{tilde over (κ)}), u=(T−T_(M))/(L_(V)/c_(p)), by resealing the above equations, we obtain the dimensionless form:

$\begin{matrix} {{ɛ\; \tau \frac{\partial\varphi}{\partial\overset{\sim}{t}}} = {{{ɛ^{2}\left( {1 + {\alpha \; u}} \right)}{{\overset{\sim}{\nabla}}^{2}\varphi}} - {\frac{1}{2}\left( {1 + {\beta \; u}} \right){g^{\prime}(\varphi)}} + {\lambda \; {{up}^{\prime}(\varphi)}}}} & (3) \\ {{\frac{\partial u}{\partial\overset{\sim}{t}} + {{r^{\prime}(\varphi)}\frac{\partial\varphi}{\partial\overset{\sim}{t}}} + {\frac{1}{2}\delta \; {g^{\prime}(\varphi)}\frac{\partial\varphi}{\partial\overset{\sim}{t}}}} = {{\overset{\sim}{\nabla}{\cdot \left\lbrack {{k(\varphi)}{\overset{\sim}{\nabla}\; u}} \right\rbrack}} + {ɛ^{2}v\frac{\partial\varphi}{\partial\overset{\sim}{t}}{{\overset{\sim}{\nabla}}^{2}\varphi}}}} & (4) \end{matrix}$

where the macroscopic length scale is R, the thermal timescale is R²/ κ, the temperature scale is L_(V)/c_(p). κ=(κ_(L)+κ_(S))/2 is the average of the thermal diffusivities κ_(L)=k_(L)/c_(p) and κ_(S)=k_(S)/c_(p) in the liquid and solid, respectively.

In the above equations, the following new parameters were used:

${ɛ = \frac{l}{R}},{\lambda = \frac{l}{6l_{c}}},{\tau = \frac{\overset{\_}{\kappa}T_{M}c_{P}}{6{ll}_{c}{ML}_{V}^{2}}},{l_{c} = {\frac{T_{M}\gamma}{L_{V}^{2}}c_{P}}},{l = \left( \frac{K}{W} \right)^{\frac{1}{2}}},{\gamma = {\frac{1}{6}({KW})^{\frac{1}{2}}}},{\mu = \frac{6{ML}_{V}l}{T_{M}}},{K = {K_{E} + {T_{M}K_{S}}}},{W = {W_{E} + {T_{M}W_{S}}}},{{Q(\varphi)} = \frac{k(\varphi)}{\overset{\_}{\kappa}c_{P}}},{\alpha = \frac{K_{S}\left( {L_{V}/c_{P}} \right)}{K}},{\beta = \frac{W_{S}\left( {L_{V}/c_{P}} \right)}{W}},{\delta = \frac{W_{E}}{L_{V}}},{v = {\frac{K_{E}}{L_{V}l^{2}}.}}$

It is easy to find that Q(0)=Q_(L)=κ_(L)/ κ and Q(1)=Q_(S)=κ_(S)/ κ satisfy Q_(L)+Q_(S)=2. By the thin interface asymptotic analysis¹⁶, special forms of the above parameters could be chosen as follows:

${{r(\varphi)} = {p(\varphi)}},{\frac{1}{Q(\varphi)} = {\frac{r(\varphi)}{Q_{L}} + \frac{1 - {r(\varphi)}}{Q_{S}}}},{\beta = {v = 0}},{\alpha = {{- \frac{76}{5}}{\lambda \left( \frac{\kappa_{L} - \kappa_{S}}{\kappa_{L} + \kappa_{S}} \right)}}},{\delta = \frac{\alpha}{\lambda}},{\frac{1}{M} = {\frac{c_{1}l^{2}{\overset{\_}{\kappa}\left( {L_{V}^{2}/c_{P}} \right)}}{\kappa_{L}\kappa_{S}T_{M}}\left( {1 - {4{c_{2}\left( \frac{\kappa_{L} - \kappa_{S}}{\kappa_{L} + \kappa_{S}} \right)}^{2}}} \right)}},$

where we require 0.0689<κ_(S)/κ_(L)<14.5, c₁=19/5, c₂=173/525.

l is the interface thickness which can be prescribed by ourselves. Therefore the undetermined parameters K_(E), K_(S), W_(E) and W_(S), can be related to l:

${K = {{l^{2}W_{E}} = {l^{2}L_{V}\delta}}},{\gamma = {\frac{1}{6}{lL}_{V}\delta}},{l_{c} = {\frac{T_{M}{lW}_{E}}{6{L_{V}^{2}/c_{P}}} = {\frac{T_{M}l\; \delta}{6{L_{V}/c_{P}}}.}}}$

The boundary condition is given by

$\begin{matrix} {{{{k(\varphi)}\frac{\partial T}{\partial n}} = {h\left( {T_{h} - T} \right)}}{{{for}\mspace{14mu} {v \cdot t}} \leq z \leq {{v \cdot t} + D}}{{{{and}\mspace{14mu} r} = R},}} & (5) \\ {{{k(\varphi)\frac{\partial T}{\partial n}} = {h\left( {T_{0} - T} \right)}}{{{{for}\mspace{14mu} z} < {v \cdot t}},{z > {{v \cdot t} + D}}}{{{{and}\mspace{14mu} r} = R},{z = 0},{z = L},}} & (6) \\ {{\frac{\partial\varphi}{\partial n} = 0}{{{{for}\mspace{14mu} z} = 0},{z = L},{{{and}\mspace{14mu} r} = {R.}}}} & (7) \end{matrix}$

The first two boundary conditions indicate that the two different directions of heat transfer: In the region covered by the heater, i.e., v·t≦z≦v·t+D, heat transfer occurs between the heater (with temperature T_(h)) and phosphorus (with temperature T(z,R,t) at position (z,R) on the boundary); in the other boundary region (z,R) with z<v·t or z>v·t+D and z=0 or L, heat transfer occurs between the water (with temperature T₀) and phosphorus (with temperature T(z,R,t) at position (z,R) and T(z,r,t) at position z=0 or L on the boundary). The last boundary condition

$\frac{\partial\varphi}{\partial n} = 0$

indicates mass conservation: n is the normal vector of the boundary. h is the heat transfer coefficient, T₀ is the room temperature and T_(h) is the temperature of the heater.

The four parameters, the length of the heater D, the radius of the phosphorus bar R, the temperature of the heater is T_(h), and the moving velocity of the heater v, determine the temperature distribution and the effect of phase separation and therefore can be adjusted to obtain good results of phase separation. By solving the governing equations with different sets of adjustable parameters numerically, both good and bad results can be observed.

In the dimensionless form, by noticing that {tilde over (x)}={tilde over (t)}/R and u=(T−T_(M))/(L_(V)/c_(p)), the boundary condition can be reduced to be

$\begin{matrix} {{\frac{\partial u}{\partial n} = {\frac{{hR}\left( {T_{h} - {{uL}_{V}/c_{P}} - T_{M}} \right)}{L_{V}\overset{\_}{\kappa}{Q(\varphi)}}\mspace{14mu} {in}\mspace{14mu} {the}\mspace{14mu} {heating}\mspace{14mu} {region}}},} & (8) \\ {{\frac{\partial u}{\partial n} = {\frac{{hR}\left( {T_{0} - {{uL}_{V}/c_{P}} - T_{M}} \right)}{L_{V}\overset{\_}{\kappa}{Q(\varphi)}}\mspace{14mu} {otherwise}}},} & (9) \\ {\frac{\partial\varphi}{\partial n} = 0} & (10) \end{matrix}$

By the radial symmetry, it is only needed to consider a cross section along a certain radius. The 3-d problem can be reduced to a 2-d problem with a rectangular computational domain (z,r)ε[0,L]×[0,R]. At r=0, it is needed

${{\frac{\partial\varphi}{\partial r}_{r = 0}} = 0},{{{{and}\mspace{14mu} \frac{\partial u}{\partial r}}_{r = 0}} = 0.}$

In cylindrical coordinate system, the dimensionless equations are

$\begin{matrix} {{\nabla\varphi} = \left( {\frac{\partial\varphi}{\partial r},{\frac{1}{r}\frac{\partial\varphi}{\partial\theta}},\frac{\partial\varphi}{\partial z}} \right)} \\ {{= \left( {\frac{\partial\varphi}{\partial r},0,\frac{\partial\varphi}{\partial z}} \right)},} \end{matrix}$ $\begin{matrix} {{\nabla^{2}\varphi} = {\Delta\varphi}} \\ {= {{\frac{1}{r}\frac{\partial\;}{\partial r}\left( {r\frac{\partial\varphi}{\partial r}} \right)} + {\frac{1}{r^{2}}\frac{{\partial^{2}\varphi}\;}{\partial\theta^{2}}} + \frac{{\partial^{2}\varphi}\;}{\partial z^{2}}}} \\ {= {{\frac{1}{r}\frac{\partial\varphi}{\partial r}} + \frac{{\partial^{2}\varphi}\;}{\partial r^{2}} + {\frac{{\partial^{2}\varphi}\;}{\partial z^{2}}.}}} \end{matrix}$

The explicit finite difference scheme could be applied to solve the differential equations. Suppose I+1 grid points and J+1 grid points are distributed along r-direction and z-direction, respectively. The dimensionless mesh size is uniform, i.e., Δr=R/I/R=1/I and Δz=L/I/R=L/(RI), where L is the length of the phosphorus cylinder in the computational domain. The functions u and φ evaluated at the point (r_(i),z_(j))=(iΔr,jΔz) at time t_(n)=nΔt are simply written in the form u^(n) _(i,j) and φ^(n) _(i,j). For i=1, . . . , I−1, j=1, . . . , J−1, the desired function values in the inner points are updated as follows:

$\begin{matrix} {\frac{\varphi_{i,j}^{n + 1} - \varphi_{i,j}^{n}}{\Delta \; t} = {{\frac{1}{\tau}\left( {1 + \alpha_{i,j}^{n}} \right)\begin{pmatrix} {{\frac{1}{r_{i}}\frac{\varphi_{{i + 1},j}^{n} - \varphi_{{i - 1},j}^{n}}{2\Delta \; r}} + \frac{\varphi_{{i + 1},j}^{n} - {2\varphi_{i,j}^{n}} + \varphi_{{i - 1},j}^{n}}{\left( {\Delta \; r} \right)^{2}} +} \\ \frac{\varphi_{i,{j + 1}}^{n} - {2\varphi_{i,j}^{n}} + \varphi_{i,{j - 1}}^{n}}{\left( {\Delta \; z} \right)^{2}} \end{pmatrix}} -}} \\ {{\frac{1}{ɛ^{2}\tau}\left( {{\frac{1}{2}{g^{\prime}\left( \varphi_{i,j}^{n} \right)}} - {\lambda \; u_{i,j}^{n}{p^{\prime}\left( \varphi_{i,j}^{n} \right)}}} \right)}} \\ {= A_{i,j}^{n}} \end{matrix}$ $\begin{matrix} {\frac{u_{i,j}^{n + 1} - u_{i,j}^{n}}{\Delta \; t} = {{\left( {{- {p^{\prime}\left( \varphi_{i,j}^{n} \right)}} - {\frac{1}{2}\delta \; {g^{\prime}\left( \varphi_{i,j}^{n} \right)}}} \right)A_{i,j}^{n}} + {Q^{\prime}\left( \varphi_{i,j}^{n} \right)}}} \\ {{\left( {{\frac{\varphi_{{i + 1},j}^{n} - \varphi_{{i - 1},j}^{n}}{2\Delta \; r}\frac{u_{{i + 1},j}^{n} - u_{{i - 1},j}^{n}}{2\Delta \; r}} + {\frac{\varphi_{i,{j + 1}}^{n} - \varphi_{i,{j - 1}}^{n}}{2\Delta \; z}\frac{u_{i,{j + 1}}^{n} - u_{i,{j - 1}}^{n}}{2\Delta \; z}}} \right) +}} \\ {{{Q\left( \varphi_{i,j}^{n} \right)}\begin{pmatrix} {{\frac{1}{r_{i}}\frac{u_{{i + 1},j}^{n} - u_{{i - 1},j}^{n}}{2\Delta \; r}} + \frac{u_{{i + 1},j}^{n} - {2u_{i,j}^{n}} + u_{{i - 1},j}^{n}}{\left( {\Delta \; r} \right)^{2}} +} \\ \frac{u_{i,{j + 1}}^{n} - {2u_{i,j}^{n}} + u_{i,{j - 1}}^{n}}{\left( {\Delta \; z} \right)^{2}} \end{pmatrix}}} \\ {= B_{i,j}^{n}} \end{matrix}$

The boundary values are updated by introducing ghost points out of the computational domain but next to the boundary:

${0 = {{\frac{\partial\varphi^{n}}{\partial r}_{({i,j})}} = \frac{\varphi_{{i + 1},j}^{n} - \varphi_{{i - 1},j}^{n}}{2\Delta \; r}}},{{\frac{\partial^{2}\varphi^{n}}{\partial r^{2}}_{({i,j})}} = {{\frac{\varphi_{{i + 1},j}^{n} - {2\varphi_{i,j}^{n}} + \varphi_{{i - 1},j}^{n}}{\left( {\Delta \; r} \right)^{2}}\mspace{14mu} {for}\mspace{14mu} i} = 0}},{{I.0} = {{\frac{\partial\varphi^{n}}{\partial z}_{({i,j})}} = \frac{\varphi_{i,{j + 1}}^{n} - \varphi_{i,{j - 1}}^{n}}{2\Delta \; z}}},{{\frac{\partial^{2}\varphi^{n}}{\partial z^{2}}_{({i,j})}} = {{\frac{\varphi_{i,{j + 1}}^{n} - {2\varphi_{i,j}^{n}} + \varphi_{i,{j - 1}}^{n}}{\left( {\Delta \; z} \right)^{2}}\mspace{14mu} {for}\mspace{14mu} j} = 0}},{{J.0} = {{\frac{\partial u^{n}}{\partial r}_{({i,j})}} = {{\frac{u_{{i + 1},j}^{n} - u_{{i - 1},j}^{n}}{2\Delta \; r}\mspace{14mu} {for}\mspace{14mu} i} = 0}}},{while}$ $\begin{matrix} {\frac{{hR}\left( {T_{h} - {u_{i,j}^{n}{L_{V}/c_{P}}} - T_{M}} \right)}{L_{V}\overset{\_}{\kappa}{Q\left( \varphi_{i,j}^{n} \right)}} = {\frac{\partial u^{n}}{\partial r}_{({i,j})}}} \\ {{= \frac{u_{{i + 1},j}^{n} - u_{{i - 1},j}^{n}}{2\Delta \; r}}\;} \end{matrix}$ ${{{for}\mspace{14mu} i} = I},{{pos} \leq j \leq {{pos} + D}},\begin{matrix} {\frac{{hR}\left( {T_{0} - {u_{i,j}^{n}{L_{V}/c_{P}}} - T_{M}} \right)}{L_{V}\overset{\_}{\kappa}{Q\left( \varphi_{i,j}^{n} \right)}} = {\frac{\partial u^{n}}{\partial r}_{({i,j})}}} \\ {{= \frac{u_{{i + 1},j}^{n} - u_{{i - 1},j}^{n}}{2\Delta \; r}}\;} \end{matrix}$ ${{{for}\mspace{14mu} i} = I},{{and}\mspace{14mu} {other}\mspace{14mu} {j'}s},{{{{and}\mspace{14mu} \frac{\partial^{2}u^{n}}{\partial r^{2}}}_{({i,j})}} = {{\frac{u_{{i + 1},j}^{n} - {2u_{i,j}^{n}} + u_{{i - 1},j}^{n}}{\left( {\Delta \; r} \right)^{2}}\mspace{14mu} {for}\mspace{14mu} i} = 0}},{I.\begin{matrix} {\frac{{hR}\left( {T_{0} - {u_{i,j}^{n}{L_{V}/c_{P}}} - T_{M}} \right)}{L_{V}\overset{\_}{\kappa}{Q\left( \varphi_{i,j}^{n} \right)}} = {\frac{\partial u^{n}}{\partial z}_{({i,j})}}} \\ {{= {{\frac{u_{i,{j + 1}}^{n} - u_{i,{j - 1}}^{n}}{2\Delta \; z}\mspace{14mu} {for}\mspace{14mu} j} = 0}},J,} \end{matrix}}$ ${{{{and}\mspace{14mu} \frac{\partial^{2}u^{n}}{\partial z^{2}}}_{({i,j})}} = {{\frac{u_{i,{j + 1}}^{n} - {2u_{i,j}^{n}} + u_{i,{j - 1}}^{n}}{\left( {\Delta \; z} \right)^{2}}\mspace{14mu} {for}\mspace{14mu} j} = 0}},{J.}$

Here pos describes the left end position of the heater and D=d/Δz is the number of grid points along z-direction inside the region under heating.

When the temperature at the boundary reaches the melting point, the phase transition from solid to liquid starts to happen.

Now only one parameter remains to be determined. It could be simply assumed that the interface thickness is l=0.1 min. In order to make the computation numerically stable, the time step Δt is chosen to be about 10⁻⁴ in dimensionless case and 10⁻³ second in the dimensional case.

All the parameters used in the above simulations are listed in Table 12^(20, 23, 24, 25).

TABLE 12 List of parameters Name of Parameter Notation value Melting point T_(M) 44.2° C. Initial and room temperature T₀ 25° C. Latent heat per unit volume L_(V) 9.71 × 10⁶ J/m³ Heat capacity per unit volume c_(P) 3.50 × 10⁵ J/(m³ · K) Density ρ 1823 kg/m³ Heat transfer coefficient h 200 W/(m² · K) Thermal conductivity of the k_(S) 0.235 W/(m · K) solid phosphorus Thermal conductivity of the k_(L) 0.187 W/(m · K) liquid phosphorus

Numerical Results:

Starting with room temperature T₀=25° C., numerically several issues related to the purification process were investigated and compared with the experimental results.

Quantities of concern numerically are: (1) fusion time: the time for the region under the heater to be fused, (2) the time for the fusion region to reach stable state (“stable” means the fusion region keeps the same shape and a constant velocity to move), (3) the velocity of both interface, (4) the width of fusion region at different radii, and (5) the lagging distance of both interfaces to both boundaries of the heater.

It is important to make clear how these quantities depend on temperature and moving velocity of the heater. In particular, the quantities that should be concern the most are the fusion time and lagging distance of left interface to left end of heater, since they have a great influence on how to choose the temperature of the heater and moving velocity of the heater.

In the following tests, the length of the heater D is set as 4 cm, and the Radius of the phosphors bar is set as 5 mm, unless specifically indicated.

When temperature of heater is T_(h)=50° C. and velocity is v=0.1 mm/s, the fusion time is 77.8 s. It can be seen that the melting region is entirely under the heater as the heater moves. The melting region almost keeps the same shape while moving (FIG. 4). The temperature is shown in FIG. 5, FIG. 6 shows the temperature in 2D at time t=400 s. FIG. 7 shows the moving velocity of the fusion region is exactly equal to the velocity of the heater.

As shown in FIG. 8, when v=0.2 mm/s, the fusion region will lag behind the heater. The characteristics are shown in Table 13.

TABLE 13 Characteristics when T_(h) = 50° C. and v = 0.2 mm/s Characteristics Quantity Value Fusion time 77.8 s Time for attaining stable state 204 s Velocity of left interface 0.2 mm/s Velocity of right interface 0.2 mm/s Width of fusion region at r = R 2.92 cm Width of fusion region at r = 0 2.40 cm Lagging distance of left interface 1.2 mm Lagging distance of right interface 17.2 mm

As shown in FIG. 9, when v=0.05 mm/s, the fusion region can stay in the middle of the heater. The characteristics are shown in Table 14.

TABLE 14 Characteristics when T_(h) = 50° C. and v = 0.05 mm/s Characteristics Quantity Value Fusion time 77.8 s Time for attaining stable state 146 s Velocity of left interface 0.05 mm/s Velocity of right interface 0.05 mm/s Width of fusion region at r = R 3.32 cm Width of fusion region at r = 0 3.06 cm Lagging distance of left interface −2.7 mm Lagging distance of right interface 6.7 mm

As shown in FIG. 10, when v=0.02 mm/s, the situation is even better. The characteristics are shown in Table 15.

TABLE 15 Characteristics when T_(h) = 50° C. and v = 0.02 mm/s Characteristics Quantity Value Fusion time 77.8 s Time for attaining stable state 180 s Velocity of left interface 0.02 mm/s Velocity of right interface 0.02 mm/s Width of fusion region at r = R 3.38 cm Width of fusion region at r = 0 3.10 cm Lagging distance of left interface −3.8 mm Lagging distance of right interface 5.2 mm

FIG. 11 shows that when v=0.5 min/s, the situation is not very good.

Then, as to the case that T_(h)=45° C. and v=0.05 mm/s, the fusion time is longer, i.e., 400.6 s. This can be seen in Table 16.

TABLE 16 Characteristics when T_(h) = 45° C. and v = 0.05 mm/s Characteristics Quantity Value Fusion time 400.6 s Time for attaining stable state 1000 s Velocity of left interface 0.05 mm/s Velocity of right interface 0.05 mm/s Width of fusion region at r = R 1.04 cm Width of fusion region at r = 0 0.84 cm Lagging distance of left interface −5.6 mm Lagging distance of right interface 26.0 mm

The width of the melting region is much narrower than the length of the heater.

According to FIG. 12, the result seems to be good. The temperature is shown is FIG. 13. The result is much better if the velocity is slowed down to be 0.02 mm/s (FIG. 14).

The result is expected to be better if the temperature of the heater is rising. Since for each fixed temperature of the heater, the situation is of course better if the velocity is slowed down. In each case, the numerical result was given when the velocity is as large as possible in consideration of the energy cost:

T_(h)=55° C. and v=0.05 mm/s (FIG. 15, Table 18); T_(h)=60° C. and v=0.05 mm/s (FIG. 16, Table 19); T_(h)=70° C. and v=0.02 mm/s (FIG. 17, Table 20); T_(h)=80° C. and v=0.02 mm/s (FIG. 18, Table 21).

TABLE 17 Characteristics when T_(h) = 55° C. and v = 0.05 mm/s Characteristics Quantity Value Fusion time 48.4 s Time for attaining stable state 112 s Velocity of left interface 0.05 mm/s Velocity of right interface 0.05 mm/s Width of fusion region at r = R 3.62 cm Width of fusion region at r = 0 3.40 cm Lagging distance of left interface −1.6 mm Lagging distance of right interface 4.4 mm

TABLE 18 Characteristics when T_(h) = 60° C. and v = 0.05 mm/s Characteristics Quantity Value Fusion time 36.6 s Time for attaining stable state 100 s Velocity of left interface 0.05 mm/s Velocity of right interface 0.05 mm/s Width of fusion region at r = R 3.74 cm Width of fusion region at r = 0 3.60 cm Lagging distance of left interface −0.8 mm Lagging distance of right interface 3.2 mm

TABLE 19 Characteristics when T_(h) = 70° C. and v = 0.02 mm/s Characteristics Quantity Value Fusion time 25.9 s Time for attaining stable state 80 s Velocity of left interface 0.02 mm/s Velocity of right interface 0.02 mm/s Width of fusion region at r = R 3.86 cm Width of fusion region at r = 0 3.82 cm Lagging distance of left interface −0.5 mm Lagging distance of right interface 1.3 mm

TABLE 20 Characteristics when T_(h) = 80° C. and v = 0.02 mm/s Characteristics Quantity Value Fusion time 20.9 s Time for attaining stable state 90 s Velocity of left interface 0.02 mm/s Velocity of right interface 0.02 mm/s Width of fusion region at r = R 3.94 cm Width of fusion region at r = 0 3.98 cm Lagging distance of left interface 0.2 mm Lagging distance of right interface 0.4 mm

It can be seen from above that although the fusion time is reduced as the temperature of the heater rises, the velocity has to be slowed down. The reduction in the fusion time is very slight, but the slowdown in velocity is remarkable. This will lead to more energy cost. Therefore, it is suggested that the temperature of the heater is between 45° C. and 50° C. This is consistent with the experiments.

In Table 21 the numerical results were listed when the temperature of the heater is between 45° C. and 50° C. and the velocity is fixed at 1 mm/min.

TABLE 21 Numerical results when T_(h) is between 45° C. and 50° C. Temperature of heater(° C.) Velocity (mm/min) Fusion time(s) 45 1 400.6 46 1 195.8 47 1 137.4 48 1 108.1 49 1 90.1 50 1 77.8

Of course this heating system can work if the length of heater is enlarged. But the purpose of the present invention is to save the energy. Therefore, it is worth to see whether it works if the length d decreases.

The temperature of heater was kept to be 50 degree Celsius and the situation that R=5 mm, d=1.5 cm and v=0.1 mm/s was tested. The result could be seen in FIG. 19. The core of the cylinder cannot be melted when d is reduced to be 1 cm (see FIG. 20).

FIG. 21 shows the case wherein R=1 cm, d=3 cm and v=0.05 mm/s. The core of the cylinder cannot be melted when d is reduced to be 2 cm (see FIG. 22).

FIG. 23 shows the case wherein R=1.5 cm, d=4 cm and v=0.05 mm/s. The core of the cylinder cannot be melted when d is reduced to be 3 cm (see FIG. 24).

Referring to the above results and Table 22, it could be roughly spoken that the ratio of the width of heater to the radius should be greater than 2, preferably greater than 3.

TABLE 22 Fusion time for different widths of heater and different radii at 50° C. Radius(cm) Width(cm) Fusion time(s) 0.5 1.5 133.2 0.5 2 86.6 0.5 4 77.8 1 3 376.0 1 4 303.2 1.5 4 1001.3

In the above experiments, the radius of cylinder was almost fixed to be 5 mm. It is obvious that if the radius is enlarged, for instance, to be 5 cm, the volume of phosphorus in every unit length is 100 times more and the efficiency of purification may be increased. But increasing in radius may lead to decreasing in moving velocity since the heat conduction in the cylinder will cost more time. Some numerical results are listed here to illustrate this phenomenon. First, the temperature of the heater is fixed to be 60 degree Celsius and the length of the heater is 10 cm. Two different velocities were used to move the heater. The results could be seen in FIG. 25 and FIG. 26.

When increasing the length of the heater to be 15 cm, the results could be seen in FIG. 27, FIG. 28, and the case wherein the length of the heater is 25 cm could be seen in FIG. 29.

The fusion time and admitted velocities for each length of the heater are listed in Table 23.

TABLE 23 Fusion time and velocities for different lengths of heater at a fixed radius R = 5 cm and a fixed temperature T_(h) = 60° C. Length of heater(cm) Fusion time(s) Admitted velocity(mm/s) 10 4023.14 0.02 15 3315.88 0.05 25 2656.395 0.1

The effects for R=5 mm and R=5 cm in the above numerical experiments were mainly tested. The parameters in the listed tables can be used to realize the purification in numerical sense. For some specified cases, for example, the case that the radius is 5 mm and the length of heater is 4 cm, both numerical results and chemical experiments give the same claim that the moving velocity should be around 0.1 mm/s.

In sum, if the radius of the phosphorus bar is enlarged, it should be applied a little higher temperature of the heater to be between 50° C. and 60° C., and also slowed down the velocity to be smaller than 0.1 mm/s, in order that the melting region can move together with the heater.

The experiments agree with the numerical simulations very well. As shown in Table 24, the inventors performed a series of experiments with different sets of parameters. “Pass” indicates that the melting zone formed. These parameters support the numerical results.

TABLE 24 Experimental results with different sets of parameters Velocity of Sets Radius Length of Temperature of heater No. (R/mm) heater (D/cm) heater (T_(h)/° C.) (v/mm/s) Result 1 5 4 45 0.02 Pass 3 5 4 48 0.02 Pass 4 5 4 50 0.05 Pass 5 5 4 60 0.05 Pass 6 5 4 70 0.1 Pass 7 5 4 80 0.1 Pass

In fact, the velocity can be raised up to 0.1 mm/s (=6 mm/min). The most optimized result is: (1) the range of speed of heater is 0.02 mm per second to 0.1 mm per second. (2) the diameter of the phosphorus is 10 mm, (3) the length of heater is 40 mm, (4) the temperature of heater is between 45° C.-60° C. The mathematical model and numerical simulations provide a guideline on how to choose the parameters so that the experiments can be carried out successfully. Although the purpose of the mathematical methods is to determine the parameter for successful heat conduction rather than to predict the concentration of As in phosphorus, it establishes a fundament for this invention. For obtaining a desired As concentration in phosphorus, the persons skilled in the art are able to repeat the running of the heater. All of these fall within the scope of the present invention.

Although the above description exemplifies few embodiments of the present invention, it should be understood that various omissions, substitutions, and changes in the form of the detail of the apparatus, system, and/or method as illustrated as well as the uses thereof, may be made by those skilled in the art, without departing from the spirit of the present invention.

CITED REFERENCES

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1. A method of production of ultrapure phosphorus by zone melting in a non-flammable environment, comprising the steps of: a. Fixing a heater in a non-flammable environment; b. Fixing a phosphorus bar through the heater; c. Setting up a motor to control the heater moving along the phosphorus bar; d. Heating a part of the said bar to become a heat zone e. Moving the heater along the said bar in order to have the meat zone moving with the heater; f. Repeat step e if necessary.
 2. The method of production of ultrapure phosphorus as mentioned in claim 1, wherein the length of the heater D, the radius of the phosphorus bar R, the temperature of the heater T_(h), and the moving velocity of the heater v shall control the solutions of the following governing equations: In r<R and 0<z<L, $\begin{matrix} {{\frac{1}{M}\frac{\partial\varphi}{\partial t}} = {{\left( {K_{E} + {TK}_{S}} \right){\nabla^{2}\varphi}} - {\frac{1}{2}\left( {W_{E} + {TW}_{S}} \right){g^{\prime}(\varphi)}} + {\frac{L_{V}\left( {T - T_{M}} \right)}{T_{M}}{p^{\prime}(\varphi)}}}} \\ {{{c_{P}\frac{\partial T}{\partial t}} + {L_{V}{r^{\prime}(\varphi)}\frac{\partial\varphi}{\partial t}} + {\frac{1}{2}W_{E}{g^{\prime}(\varphi)}\frac{\partial\varphi}{\partial t}}} = {{\nabla{\cdot \left\lbrack {{k(\varphi)}{\nabla\; T}} \right\rbrack}} + {K_{E}\frac{\partial\varphi}{\partial t}{\nabla^{2}\varphi}}}} \end{matrix}$ The boundary conditions are: ${{{k(\varphi)}\frac{\partial T}{\partial n}} = {h\left( {T_{h} - T} \right)}},{{{for}\mspace{14mu} {v \cdot t}} \leq z \leq {{v \cdot t} + D}}$ ${{{and}\mspace{14mu} r} = R},{{{k(\varphi)}\frac{\partial T}{\partial n}} = {h\left( {T_{0} - T} \right)}},{{{for}\mspace{14mu} z} < {v \cdot t}},{z > {{v \cdot t} + D}}$ ${{{and}\mspace{14mu} r} = R},{z = 0},{z = L},{\frac{\partial\varphi}{\partial n} = 0},{{{for}\mspace{14mu} r} = R},{z = 0},{{z = L};}$ In these equations, φ(x,t) is an continuous function which labels the two phases at the position x and the time t, with φ=0 in the solid phase, φ=1 in the liquid phase and 0<0<1 in the interfacial region; M is the mobility coefficient, K_(E) and K_(S) are interfacial energy coefficients, W_(E) and W_(S) are barrier heights for double well potential function g(φ)=φ²(1−φ)²; p(φ)=φ²(3−2φ); L_(V); is the latent heat per unit volume and T_(M) is the melting point; T(x,t) is the temperature distribution at the position x and the time t; c_(p), is the heat capacity per unit volume, r(φ)=p(φ); k(φ) permits different thermal conductivities k_(L)=k(0) and k_(S)=k(1) in the bulk liquid and solid phases, respectively; these governing equations being solved on a cylindrical domain x=(z,r)ε[0,L]×[0,R], L is the length of phosphorus cylinder and R is the radius, z is the axial direction coordinate, r is the radial direction coordinate, D is the length of the heater, n is the outer normal vector of the boundary, h is the heat transfer coefficient, T₀ is the room temperature and T_(h) is the temperature of the heater.
 3. The method of production of ultrapure phosphorus as mentioned in any one of the preceding claims, wherein the said phosphorus bar can be molded by a plastic mold, preferably a high density polyethylene or Teflon tube with a round hole in the middle.
 4. The method of production of ultrapure phosphorus as mentioned in any one of the preceding claims, where the length of the heater is at least 2 times, preferably at least 3 times, more preferably at least 4 times, longer than the diameter of the phosphorus bar.
 5. The method of production of ultrapure phosphorus as mentioned in any one of the preceding claims, wherein the temperature of heater is between 45° C. and 80° C.
 6. The method of production of ultrapure phosphorus as mentioned in any one of the preceding claims, wherein the temperature of heater is between 45° C. and 60° C.
 7. The method of production of ultrapure phosphorus as mentioned in any one of the preceding claims, wherein the temperature of heater is between 45° C. and 50° C.
 8. The method of production of ultrapure phosphorus as mentioned in any one of the preceding claims, wherein the velocity of the heater is between 0.02 mm/s and 0.1 mm/s.
 9. The method of production of ultrapure phosphorus as mentioned in any one of the preceding claims, wherein the velocity of the heater is between 0.05 mm/s and 0.1 mm/s.
 10. A apparatus for the production of ultrapure phosphorus as mentioned in the preceding claims, which comprises in a non-flammable environment: g. heater; h. a phosphorus bar fixed through the heater and between fixing units; i. a motor set up to control the heater moving along the phosphorus bar. 